On the $m$th order $p$-affine capacity

TL;DR

The work develops a comprehensive theory of the m-th order $p$-affine capacity $C_{p,Q}(K)$ for compact sets in $M_{n,1}(\mathbb{R})$, extending affine isoperimetric/isocapacity frameworks to higher-order projections. Central contributions include multiple equivalent definitions, invariance and regularity properties, exact ball values, and tight comparisons with the $p$-variational capacity and volume, as well as links to the $m$th order $p$-integral affine surface area $\Phi_{p,Q}(K)$ and the $L_p$ surface area $S_p(K)$ via the Orlicz/projection-body machinery. The core results establish a consistent chain of affine inequalities among volume, $C_{p,Q}$, $\Phi_{p,Q}$, and $S_p$, with ellipsoids (and in particular origin-symmetric ellipsoids) as equality cases, thus strengthening the affine viewpoint in the higher-order setting. These findings provide a robust framework for higher-order affine Sobolev-type inequalities and related geometric-analytic problems.

Abstract

Let $M_{n, m}(\mathbb{R})$ denote the space of $n\times m$ real matrices, and $\mathcal{K}_o^{n,m}$ be the set of convex bodies in $M_{n, m}(\mathbb{R})$ containing the origin. We develop a theory for the $m$th order $p$-affine capacity $C_{p,Q}(\cdot)$ for $p\in[1,n)$ and $Q\in\mathcal{K}_{o}^{1,m}$. Several equivalent definitions for the $m$th order $p$-affine capacity will be provided, and some of its fundamental properties will be proved, including for example, translation invariance and affine invariance. We also establish several inequalities related to the $m$th order $p$-affine capacity, including those comparing to the $p$-variational capacity, the volume, the $m$th order $p$-integral affine surface area, as well as the $L_p$ surface area.

Theorems & Definitions (31)

• Definition 3.1
• Proposition 3.2
• proof
• Theorem 3.3
• proof
• Corollary 3.4
• proof
• Proposition 3.5
• proof
• Proposition 3.6